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- * Olympiad is recommended for high school students who are already studying math at an ...combinatorics, and number theory, along with sets of accompanying practice problems at the end of every section.23 KB (3,038 words) - 18:33, 15 February 2025
- ...pon a variable. However, when multiplying/dividing both sides by negative numbers, we have to flip the sign. A more complex example is <math>\frac{x-8}{x+5}+4\ge 3</math>.12 KB (1,806 words) - 05:07, 19 June 2024
- ...y-Schwarz forms the foundation for inequality problems in intermediate and olympiad competitions. It is particularly crucial in proof-based contests. ...ion comes in handy in linear algebra problems at intermediate and olympiad problems.13 KB (2,048 words) - 14:28, 22 February 2024
- ...ts argument is nonnegative. The range is the set of all non-negative real numbers, because the square root can never return a negative value. ...lly we speak about functions whose domain is also a [[subset]] of the real numbers.10 KB (1,761 words) - 02:16, 12 May 2023
- ...e to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ven precision can allow easier comparison than computing and comparing the numbers themselves. Logs also allow (with repetition) to turn left to right exponen4 KB (680 words) - 11:54, 16 October 2023
- The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math> ...er]] cannot be [[negative]], so this equation has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <5 KB (860 words) - 14:36, 10 December 2023
- ...Any [[complex number]] can be expressed as <math>a+bi</math> for some real numbers <math>a</math> and <math>b</math>. ==Problems==2 KB (321 words) - 14:57, 5 September 2008
- ...finition of trigonometry is preferred in order to extend trigonometry to a complex domain. ...d angles. As such, this definition is usually preferred in intermediate to olympiad geometry settings.8 KB (1,228 words) - 14:40, 10 January 2025
- Variables are very useful in solving many problems [[algebra]]icly. For instance, [[word problem]]s can often be solved using ...w, \omega</math> and <math>\zeta</math> are often used as [[complex number|complex]] variables. <math>m</math> and <math>n</math> are often used as variables1 KB (203 words) - 20:35, 15 November 2007
- import olympiad; ...e locus of the midpoints of <math>AD</math> is clearly a circle. Since the problems gives that <math>AD</math> is the only chord starting at <math>A</math> bis20 KB (3,497 words) - 14:37, 27 May 2024
- We will give a solution using complex coordinates. The first step is the following lemma. ...x</math>, <math>y</math> and <math>z</math> are complex. The circle in the complex plane passing through <math>x</math>, <math>x + ty</math> and <math>x + (s5 KB (986 words) - 21:46, 18 May 2015
- * Intermediate (at the level of hardest [[AMC 12]] problems, the [[AIME]], [[ARML]], and the [[Mandelbrot Competition]]). * [[Intermediate Trigonometry/Complex Numbers Course]] [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#in2 KB (303 words) - 15:02, 11 July 2006
- * [[Complex number | Complex numbers]] * [[Word problem | Word problems]]991 bytes (86 words) - 14:58, 24 August 2024
- ...tinct [[complex number]]s <math> x_0, \ldots , x_n </math> and any complex numbers <math> y_0, \ldots, y_n </math>, there exists a unique [[polynomial]] <math This formula is useful for many olympiad problems, especially since such a polynomial is unique.2 KB (398 words) - 02:50, 20 November 2023
- ...</math>. Suppose that <math>|P(i)| < 1</math>. Prove that there exist real numbers <math>a</math> and <math>b</math> such that <math>P(a + bi) = 0</math> and ...ine{w}) = \overline{ P(w)} = 0</math>, so <math>\overline{w}</math>, the [[complex conjugate]] of <math>w</math>, is also a root of <math>P</math>.2 KB (340 words) - 18:11, 18 July 2016
- ...d in the form <math>T(x) = x_0+k (x-x_0),</math> where <math>k</math> is a complex number. The magnitude <math>|k|</math> is the dilation factor of the spiral ...defined by the images of two distinct points. It is easy to show using the complex plane.28 KB (4,687 words) - 21:27, 19 February 2025
- Problems from the '''1973 [[United States of America Mathematical Olympiad | USAMO]]'''. [[1973 USAMO Problems/Problem 1 | Solution]]2 KB (273 words) - 17:53, 3 July 2013
- ...h>\, 1992 \,</math> and has distinct zeros. Prove that there exist complex numbers <math>\, a_1, a_2, \ldots, a_{1992} \,</math> such that <math>\, P(z) \,</m ..._{1}, z_{2}, \dots, z_{1992}</math>, there exist <math>1992</math> complex numbers <math>a_{1}, a_{2}, \ldots, a_{1992}</math>, such that the polynomial5 KB (825 words) - 01:21, 2 September 2016
- Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to train students for future [[mathematics]] competitions. ...93238&sr=1-1-catcorr Advanced Olympiad Inequalities: Algebraic & Geometric Olympiad Inequalities] by '''Alijadallah Belabess'''.19 KB (2,581 words) - 13:25, 1 November 2024
- ...tiple of <math>\pi</math>. <math>x, y, </math> and <math>z</math> are real numbers. If <math>x\sin(A)+y\sin(B)+z\sin(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0< ...}</math>, <math>b=ye^{iB}</math>, <math>c=ze^{iC}</math> be numbers in the complex plane.2 KB (347 words) - 10:20, 6 May 2017